The structural assumption and completeness assumption including observable confounders are given as follows.
[Structural Assumption [1]}] 
We assume \(Y Z | A, U, X\) , and \(W (A,Z) | U, X\) .

[Completeness Assumption [1]}] 
Let \(l: \mathcal {U} \rightarrow \mathbb {R}\)  be any square integrable function \(\Vert l\Vert _{{U}}\) . We assume the following:

l(U) A=a, Z=z, X=x=0   (a,z,x) A Z X       l(u) = 0 a.s.
